Treewidth

1. Treewidth Algorithms and Networks

2. Overview • Historic introduction: Series parallel graphs • Dynamic programming on trees • Dynamic programming on series parallel graphs • Treewidth • Dynamic programming on graphs of small treewidth • Finding tree decompositions Treewidth

3. R1 R2 R1 R2 Computing the Resistance With the Laws of Ohm 1789-1854 Two resistorsin series Two resistorsin parallel Treewidth

4. Repeated use of the rules 6 6 5 2 2 1 7 Has resistance 4 1/6 + 1/2 = 1/(1.5) 1.5 + 1.5 + 5 = 8 1 + 7 = 8 1/8 + 1/8 = 1/4 Treewidth

5. A tree structure 5 P 7 6 2 S S P P 5 7 1 1 6 2 6 2 6 2 Treewidth

6. 5 7 6 2 4 1 6 2 Carry on! • Internal structure of graph can be forgotten once we know essential information about it! ¼ + ¼ = ½ Treewidth

7. Using tree structures for solving hard problems on graphs 1 • Network is ‘series parallel graph’ • 196*, 197*: many problems that are hard for general graphs are easy for • Trees • Series parallel graphs • Many well-known problems e.g.: NP-complete Linear / polynomial time computable Treewidth

8. Weighted Independent Set • Independent set: set of vertices that are pair wise non-adjacent. • Weighted independent set • Given: Graph G=(V,E), weight w(v) for each vertex v. • Question: What is the maximum total weight of an independent set in G? • NP-hard Treewidth

9. Weighted Independent Set on Trees • On trees, this problem can be solved in linear time with dynamic programming. • Choose root r. For each v, T(v) is subtree with v as root. • Write A(v) = maximum weight of independent set S in T(v) B(v) = maximum weight of independent set S in T(v), such that vÏ S. Treewidth

10. Recursive formulations • If v is a leaf: • A(v) = w(v) • B(v) = 0 • If v has children x1, … , xr: A(v) = max{ w(v) + B(x1) + … + B(xr) , A(x1) + … A(xr) } B(v) = A(x1) + … A(xr) Treewidth

11. Linear time algorithm • Compute A(v) and B(v) for each v, bottom-up. • E.g., in postorder • Constructing corresponding sets can also be done in linear time. Treewidth

12. Second example:Weighted dominating set • A set of vertices S is dominating, if each vertex in G belongs to S or is adjacent to a vertex in S. • Problem: given a graph G with vertex weights, what is the minimum total weight of a dominating set in G? • Again, NP-hard, but linear time on trees. Treewidth

13. Subproblems • C(v) = minimum weight of dominating set S of T(v) • D(v) = minimum weight of dominating set S of T(v) with vÎ S. • E(v) = minimum weight of a set S of T(v) that dominates all vertices, except possibly v. Treewidth

14. Recursive formulations • If v is a leaf, … • If v has children x1, … , xr: • C(v) = the minimum of: • w(v) + E(x1) + … + E(xr) • C(x1) + … + C(xi-1) + D(xi) + C(xi+1) + … + C(xr), over all i, 1 £i£ r. • D(v) = w(v) + E(x1) + … + E(xr) • E(v) = min { w(v) + E(x1) + … + E(xr), C(x1) + … + C(xr) } Treewidth

15. Gives again a linear time algorithm • Compute bottom up (e.g., postorder), and use another type of dynamic programming for the values C(v). • Constructing sets can also be done in linear time Treewidth

16. Generalizing to series parallel graphs • A 2-terminal graph is a graph G=(V,E) with two special vertices s and t, its terminals. • A 2-terminal (multi)-graph is series parallel, when it is: • A single edge (s,t). • Obtained by series composition of 2 series parallel graphs • Obtained by parallel composition of 2 series parallel graphs Treewidth

17. Series composition s1 s1 s2 + s2=t1 s2 t1 t2 t2 Treewidth

18. Parallel composition s1 s2 s1 s1=s2 + t1 t2 t1 t1=t2 Treewidth

19. Series Parallel Graphs have an SP-tree 5 P 7 6 2 S S P P 5 7 1 1 6 2 6 2 6 2 Treewidth

20. G(i) 5 • Associate to each node i of SP tree a 2-terminal graph G(i). 6 2 P S S 6 2 P P 5 7 1 6 2 6 2 Treewidth

21. Maximum weighted independent set for series parallel graphs • G(i), say with terminals s and t • AA(i) = maximum weight of independent set S of G(i) with s Î S, t Î S • BA(i) = maximum weight of independent set S of G(i) with sÏ S, tÎ S • AB(i) = maximum weight of independent set S of G(i) withsÎ S, tÏ S • BB(i) = maximum weight of independent set S of G(i) with sÏ S, tÏ S Treewidth

22. Maximum weighted independent set of series parallel graphs 2 • Computing AA, AB, BA, BB for • Leaves of SP-tree: trivial • Series, parallel composition: case analysis, using values for sub-sp-graphs G(i1), G(i2) • E.g., series operation, s’ terminal between i1 and i2 • AA(i) = max {AA(i1)+AA(i2) – w(s’), AB(i1) + BA(i2) } • O(1) time per node of SP-tree: O(n) total. Treewidth

23. Many generalizations • Many other problems • Other classes of graphs to which we can assign a tree-structure, including • Graphs of treewidth k, for small k. Treewidth

24. Tree decomposition g a • A tree decomposition: • Tree with a vertex set associated to every node. • For all edges {v,w}: there is a set containing both v and w. • For every v: the nodes that contain v form a connected subtree. h b c f e d a a g c f a g h b c f d c e Treewidth

25. Tree decomposition g a • A tree decomposition: • Tree with a vertex set associated to every node. • For all edges {v,w}: there is a set containing both v and w. • For every v: the nodes that contain v form a connected subtree. h b c f e d a a g c f a g h b c f d c e Treewidth

26. Treewidth (definition) g a h • Width of tree decomposition: • Treewidth of graph G: tw(G)= minimum width over all tree decompositions of G. b c f e d a a g c f a g h b c f d c e a b c d e f g h Treewidth

27. Some graphs have small treewidth • Appearing in some applications (e.g., probabilistic networks) • Trees have treewidth 1 • Series Parallel graphs have treewidth 2. • … Treewidth

28. Choose a root r Take Xr = {r}, and for each other node i:Xi= {i, parent(i)} T with these bags gives a tree decomposition of width 1 Trees have treewidth one a b e a c d b a e a c b d b Treewidth

29. Algorithms using tree decompositions • Step 1: Find a tree decomposition of width bounded by some small k. • Discussed later • Step 2. Use dynamic programming, bottom-up on the tree. Treewidth

30. Determining treewidth • Treewidth problem (decision version): • Given: Graph G, integer k • Question: Is the treewidth of G at most k • Treewidth problem (construction version): • Given: Graph G • Question: construct a tree decomposition of G with minimum width • NP-complete (Arnborg, Proskurowski) • (n) time algorithm for fixed k (B.) • Practical O(n) algorithms for k= 1, 2, 3 (Arnborg, Corneil, Proskurowski), k = 4 (Sanders; Hein and Koster) • Practical O*(2n) algorithm for small graphs (B. et al.) • Many (often good) heuristics • Sometimes by construction, or construction comes from application (trees, sp-graphs, planar graphs with small diameter, …) Treewidth

31. Separator property w i If both v and w not in Xi, then v and w are not adjacent v Treewidth

32. Nice tree decompositions • Rooted tree, and four types of nodes i: • Leaf: leaf of tree with |Xi| = 1. • Join: node with two children j, j’ with Xi= Xj = Xj’. • Introduce: node with one child j with Xi = XjÈ {v} for some vertex v • Forget: node with one child j with Xi= Xj- {v} for some vertex v • There is always a nice tree decomposition with the same width. Treewidth

33. Define G(i) • Nice tree decomposition. • For each node i, G(i) subgraph of G, formed by all nodes in sets Xj, with j=i or j a descendant of i in tree. • Notate: G(i) = ( V(i), E(i) ). Treewidth

34. Maximum weighted independent set on graphs with treewidth k • For node i in tree decomposition, SÍ Xi write • R(i, S) = maximum weight of independent set W of G(i) with WÇ Xi = S, • – ¥ if such W does not exist Treewidth

35. Leaf nodes • Let i be a leaf node. Say Xi = {v}. • R(i,{v}) = w(v) • R(i, Æ ) = 0 v G(i) is a graph with one vertex Treewidth

36. Join nodes • Let i be a join node with children j1, j2. • R(i, S) = R(j1, S) + R(j2, S) – w(S). = + Treewidth

37. Introduce nodes • Let i be a node with child j, with Xi = Xj È {v}. • Let S Í Xj. • R(i,S) = R(j,S). • If v not adjacent to vertex in S: R(i,SÈ{v})=R(j,S) + w(v) • If v adjacent to vertex in S: R(i,S È {v}) = – ¥. v Treewidth

38. Forget nodes • Let i be a node with child j, with Xi = Xj- {v}. • Let S Í Xi. • R(i,S) = max (R(j,S), R(j,S È {v})) v v Treewidth

39. Maximum weighted independent set on graphs with treewidth k • For node i in tree decomposition, SÍ Xi write • R(i, S) = maximum weight of independent set W of G(i) with WÇ Xi = S, – ¥ if such W does not exist • Compute for each node i, a table with all values R(i, …). • Each such table can be computed in O(2k) time when treewidth at most k. • Gives O(n) algorithm when treewidth is (small) constant. Treewidth

40. Frequency assignment problem • Given: • Graph G=(V,E) • Frequency set F(v) ÍN for all vÎV • Cost function • c(e,r,s) , e = {v,w}, r a frequency of v, s a frequency of w • Question • Find a function g with • For all v Î V: g(v) Î F(v) • The total sum over all edges e={v,w} of c(e,g(v),g(w)) is as small as possible Treewidth

41. Frequency assignment when treewidth is small • Suppose sets F(v) are small • Suppose G has small treewidth • Algorithm exploits tree decomposition What tables are we computing? • Leaf: trivial • Introduce: … • Forget: projection • Join: sum but subtract double terms Treewidth

42. General method • Compute a tree decomposition • E.g., with minimum degree heuristic • Make it nice • Use dynamic programming • Works for many problems • Courcelle: those that can be formulated in monadic second order logic • Practical: TSP, frequency assignment, problems on planar graphs like dominating set, probabilistic inference Treewidth

43. A lemma • Let ({Xi | iÎ I},T) be a tree decomposition of G. Let Z be a clique in G. Then there is a jÎ I with Z Í Xj. • Proof: Take arbitrary root of T. For each vÎ Z, look at highest node containing v. Look at such highpoint of maximum depth. Treewidth

44. The minimum degree heuristic A heuristic for treewidth Works often well • Repeat: • Take vertex v of minimum degree • Make neighbors of v a clique • Remove v, and repeat on rest of G • Add v with neighbors to tree decomposition N(v) N(v) vN(v) Treewidth

45. Other heuristics • Minimum fill-in heuristic • Similar to minimum degree heuristic, but takes vertex with smallest fill-in: • Number of edges that must be added when the neighbours of v are made a clique • Other choices of vertices, refining, using separators, … Treewidth

46. Representation as permutation • A correspondence between tree decompositions and permutations of the vertices • Repeat: remove superfluous leaf bag, or take vertex that appears in 1 leaf bag and no other bag • Make neighbours of v = p(1) into a clique; recursively make tree decomposition of graph – v; add bag with v and neighbours • Used in heuristics, and local search methods (e.g., taboo search, simulated annealing) and genetic algorithms Treewidth

47. Connection to Gauss eliminating • Consider Gauss elimination on a symmetric matrix • For n by n matrix M, let GM be the graph with n vertices, and edge (i,j) if Mij¹ 0 • If we eliminate a row and corresponding column, effect on G is: • Make neighbors of v a clique • Remove v Treewidth

48. Application: Probabilistic networks • Lauritzen-Spiegelhalter algorithm for inference on probabilistic networks (belief networks) uses a tree decomposition of the moralized form of the network • Underlying several modern decision support networks Treewidth

49. Treewidth

50. Designing a DP algorithm • Methodology: • What are “partial certificates”? • What characterizes a partial certificate (essential for extending to full certificate)? Gives set of subproblems • Give recurrences for subproblems • Find order in which recurrences are evaluated; or use memorization • Give algorithm; possibly save memory or make construction version Treewidth